A Bose-Hubbard Hamiltonian, modeling cold bosons in an optical lattice, is used to simulate the dynamics of interacting open quantum systems as subsystems a larger closed system, avoiding complications like the introduction of baths, complex absorbing potentials or absorbing boundaries. The numerically exact unitary dynamics is compared with effective descriptions of the subsystems based on non-Hermitian Hamiltonians or Lindblad master equations. The validity of popular models with constant decay rates is explicitly analyzed for decaying single and double wells. In addition we present a discrete lattice version of the Siegert approximation method for calculating decay rates.
A Bose-Hubbard Hamiltonian, modeling cold bosons in an optical lattice, is used to simulate the dynamics of interacting open quantum systems as subsystems a larger closed system, avoiding complications like the introduction of baths, complex absorbing potentials or absorbing boundaries. The numerically exact unitary dynamics is compared with effective descriptions of the subsystems based on non-Hermitian Hamiltonians or Lindblad master equations. The validity of popular models with constant decay rates is explicitly analyzed for decaying single and double wells. In addition we present a discrete lattice version of the Siegert approximation method for calculating decay rates.
In Chapter 1 we set up the physical and mathematical framework for the description of interacting
bosons on a lattice. After a brief introduction to the basic concepts of Bose-Einstein condensation, and
how the condensate can be manipulated using optical lattices, we derive the Bose-Hubbard Hamiltonian
(BHH) which is the paradigm model as far as the quantum description is concerned. Then the semiclassical
limit of the BHH – the discrete nonlinear Schroedinger equation (DNLS) – as well as the mean-field
(classical) description, the so-called Gross-Pitaevskii equation (GPE), is discussed. We give an overview
of the relevant literature and of the various physical systems that are captured by the BHH including,
among others, arrays of Josephson junctions and bond vibrations in small molecules. The chapter ends
with an outline of the remaining thesis.
In Chapter 2, we look at some interesting experiments done in cold bosonic systems. Ostensibly,
done to realize the dc and ac Josephson effect in a system of cold bosons, it was soon found that the
system can show some rather interesting non-linear behavior. This includes features such as solitons
and breathers , but most interestingly, it shows the damping of the center-of-mass motion. Theoretical
explanations are given for both the experiments. The damping of center-of-mass of the system is seen as
a result of modulational instability. We give a different interpretation to this opening up lines for further
investigations. We outline these at the end of the chapter.
In Chapter 3, we discuss various experiments which contradicted the idea of a sharp modulational
instability. Rather, the loss of coherence is smeared out over a large domain. We then reproduce some
calculations from the literature showing how the dissipation can be thought of as arising from quantum
phase slips. We calculate the tunneling rates for quantum phase slips and show a phase diagram demarcating
stable and unstable regions of current which have been have obtained from mean-field calculations
via the Gutzwiller ansatz.
In Chapter 4, we look at the problem in details analyzing it as a Tomonaga-Luttinger liquid. The
framework of Tomonaga-Luttinger liquid is actually a non-perturbative tool through which we can investigate
an one-dimensional interacting bosonic fluid. We look at two possible mechanisms of decay
of current- We look at the effect of a weak link in a Josephson Junction array as well as local quantum
phase slips on a bosonic superfluid. We find that quantum phase slips are actually irrelevant under
renormalisation group in both cases and the supercurrent is revived.
In chapter 5, we look at the non-linear dynamics of the structures in greater detail, thus elucidating
chaos. This mechanism of chaos is different from all realizations of chaos known so far and is a result of
phase fluctuations, which are encoded by quantum phase slips.
We study the full-fledged microscopic dynamics of two interacting, ultracold bosons in a one-dimensional double-well potential, through the numerically exact diagonalization of the many-body Hamiltonian. With the particles initially prepared in the left well, we increase the width of the right well in subsequent trap realizations and witness how the tunneling oscillations evolve into particle loss. In this closed system, we analyze the spectral signatures of single- and two-particle tunneling for the entire range of repulsive interactions. We conclude that for comparable widths of the two wells, pairwise tunneling of the bosons may be realized for specific system parameters. In contrast, the decay process (corresponding to a broad right well) is dominated by uncorrelated single-particle decay.
Beautiful experiments have shown that bosons in optical lattices provide a physical system with an unprecedented degree of control; that they can generate strongly correlated states; and that, as long-term goals, they may be useful for simulating other many-body systems, or realising a quantum computer. Recently a practical scheme for measuring temperature in an optical lattice has been demonstrated, which marks an important milestone necessary for the detailed study of the thermodynamics of this system. In this thesis, we consider both the translationally-invariant lattice, and the combined harmonic trap and optical lattice. We use an extended Bose-Hubbard Hamiltonian which goes beyond the usual Bose-Hubbard approach, and is valid for shallower lattices and higher temperatures, by allowing for beyond nearest-neighbour hopping and excited bands, and we have developed an approximation scheme for off-site interactions. We derive the equations of the Popov approximation to the Hartree-Fock-Bogoliubov method for our Hamiltonian, and diagonalise these equations in the local density approximation (LDA). We examine the density of states of the optical lattice in detail and in various limits. We derive new results on the structure of the density of states, and, in the ideal case, we compare the density of states from the full diagonalisation with our LDA calculation. We make an efficient numerical implementation of our theory and compare the results with the full diagonalisation (for the non-interacting case) and with the limited experimental results currently available. We consider the significance of beyond nearest-neighbour hopping and excited bands and illustrate the properties of our model. In contrast to the trapped gas with no lattice, few thermodynamic results for cold bosons in an optical lattice have been calculated. We analytically derive the first practical formula for the critical temperature in an optical lattice by using simplified shapes for the density of states. We derive corrections for the influence of excited bands, the finite-size effect and mean-field interactions. In all of these cases, we compare our results to full numerical calculations and show that the validity range of our method is complementary to that of the effective-mass approximation, so that the simple descriptions extend over a wide range.
The realization of Bose-Einstein condensation in dilute atomic gases opens an exciting way to quantum mechanics and begins a new area of quantum simulation. As a macroscopic quantum object and a many-body bosonic system, the Bose-Einstein condensates can show numerous exotic quantum effects and have naturally attracted great attention. One of the simplest quantum many-body systems to be realized experimentally and studied theoretically is ultra-cold atoms in a double-well potential. This system can exhibit a great variety of quantum interference phenomena such as tunneling oscillation, self-trapping and the entanglement of macroscopic superpositions. Specifically, the double-well potentials built by optical or magnetic fields are easy to change and the many-body interaction between ultra-cold atoms can be changed by the method of Feshbach resonance, enabling the precise quantum control of the double-well dynamics of the condensates. In the present work, we study the dynamics of a condensate in a trapping potential consisting of an unalterable double-well trap and an additional moving optical lattice. If the lattice space is much smaller than the size of the double-well trap, the system can be simplified into a double-well trapped condensate with a tunable effective mass. Using the mean-field factorization assumption, together with a two-mode approximation, we obtain the analytic expressions for the dependence of the tunneling rate and the self-collision strength on the effective mass. The tunneling rate decays and the collision strength grows up with the increase of the effective mass. As a consequence of their different changes, we conclude that the adjustment of the effective mass of the ultra-cold atoms, rather than the changing of the trap barrier or adjusting of the atomic scattering length, is an alternative approach to controlling the double-well dynamics of the condensate. Via numerical simulations of the mean-field dynamical equations with some realistic parameters, we show that a transition between the quantum coherent tunneling and the self-trapping behaviors is experimentally realizable with the mass-control approach. Specifically, we show that the approach is still valid for the case of negative mass. Moreover, we find that the negative-mass case can be used even to stimulate the double-well dynamics of the condensate with a negative atomic scattering length.
We present the Redfield quantum master equation (RQME) description for an open system of non-interacting particles (bosons or fermions) on an arbitrary lattice of N sites in any dimension and connected to multiple reservoirs at different temperatures and chemical potentials. For the N=2 case, we show that RQME gives results which agree with exact analytical results for steady state properties and with exact numerics for time-dependent properties. In comparision, we show that the Lindblad equation approach has a limited domain of validity. These results can be experimentally relevant for cold atoms, cavity QED and far-from-equilibrium quantum dot experiments.
Experiments with ultracold atoms in optical lattice have become a versatile testing ground to study diverse quantum many-body Hamiltonians. A single-band Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in 1998 and its associated quantum phase-transition was subsequently observed in 2002. Over the years, there has been a rapid progress in experimental realizations of more complex lattice geometries, leading to more exotic BH Hamiltonians with contributions from excited bands, and modified tunneling and interaction energies. There has also been interesting theoretical insights and experimental studies on “un- conventional” Bose-Einstein condensates in optical lattices and predictions of rich orbital physics in higher bands. In this thesis, I present our results on several multi- band BH models and emergent quantum phenomena. In particular, I study optical lattices with two local minima per unit cell and show that the low energy states of a multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec- tive single-band Hamiltonian with strong three-body interactions. I also propose a second method to create three-body interactions in ultracold gases of bosonic atoms in a optical lattice. In this case, this is achieved by a careful cancellation of two contributions in the pair-wise interaction between the atoms, one proportional to the zero-energy scattering length and a second proportional to the effective range. I subsequently study the physics of Bose-Einstein condensation in the second band of a double-well 2D lattice and show that the collision aided decay rate of the con- densate to the ground band is smaller than the tunneling rate between neighboring unit cells. Finally, I propose a numerical method using the discrete variable repre- sentation for constructing real-valued Wannier functions localized in a unit cell for optical lattices. The developed numerical method is general and can be applied to a wide array of optical lattice geometries in one, two or three dimensions.